a). Find the critical points of the following functions on the domain or on the given interval.

b). Use a graphing utility to determine whether each critical point corresponds to a local minimum, local maximum, or neither
f(x)= x- tan^-1 x

f = x - arctan(x)

f' = 1 - 1/(1+x^2) = x^2/(1+x^2)

f'=0 only at x=0

f" = 2x/(1+x^2)^2
f"(0) = 0, so x=0 is not a min or a max, but a point of inflection.

See the graph at

http://www.wolframalpha.com/input/?i=x+-+arctan%28x%29

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To find the critical points of the function f(x) = x - tan^(-1)(x) and determine whether they correspond to a local minimum, local maximum, or neither, we can follow these steps:

a) Finding the Critical Points:
1. To find the critical points, we need to find the values of x where the derivative of the function is equal to zero or is undefined.

2. Start by finding the derivative of the function f(x) with respect to x. The derivative of f(x) can be found using the chain rule.

f'(x) = 1 - (1 / (1 + x^2))

3. Set the derivative equal to zero and solve for x:

1 - (1 / (1 + x^2)) = 0

Simplifying this equation, we get:

1 + x^2 = 1

x^2 = 0

Hence, x = 0 is the critical point.

4. Aside from the points where the derivative is zero, we also have to check for values of x where the derivative is undefined. In this case, the derivative is undefined when the denominator (1 + x^2) is equal to zero. However, since the denominator is always positive, there are no critical points from this condition.

Therefore, the critical point of the function f(x) = x - tan^(-1)(x) is x = 0.

b) Determining Local Minimum/Maximum:
1. To determine whether the critical point x = 0 corresponds to a local minimum, local maximum, or neither, we can utilize a graphing utility.

2. Plot the graph of the function f(x) = x - tan^(-1)(x) using an online graphing tool or mathematics software. The graph should show the behavior of the function around the critical point x = 0.

3. Examine the graph around x = 0. If the function has a local minimum at x = 0, the graph should display a downward-facing U-shape around that point. If the function has a local maximum, the graph should show an upward-facing U-shape around x = 0. If the graph is relatively flat or random near x = 0, it indicates that x = 0 is neither a local minimum nor a local maximum.

By visually analyzing the graph, we can determine whether x = 0 corresponds to a local minimum, local maximum, or neither.